The bordered-curve complement invariance conjecture for knot Floer homology

Let $K\subset Y$ be a knot with complement $M=Y\setminus\nu(K)$. The decorated curves $\widehat{\mathit{HF}}(Y,K)$ are curves in the punctured torus $T_M^*$, and $\widehat{\mathit{HF}}(M)$ denotes the decorated curves associated to the bordered Floer invariant of $M$. Complement invariance conjecture. For any knot $K\subset Y$ with complement $M=Y\setminus\nu(K)$, the decorated curves $\widehat{\mathit{HF}}(Y,K)$ in $T_M^*$ agree with the curves $\widehat{\mathit{HF}}(M)$ defined in the cited work; in particular, they are invariants of the knot complement $M$. This would identify the knot Floer construction with the bordered Floer immersed-curve invariant and show that it depends only on the complement.